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import Mathlib.Data.Real.EReal
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Bochner
import NMDSProofs.Utils
import NMDSProofs.RKHS
import NMDSProofs.PushForward
import NMDSProofs.SteepestDirection
local macro_rules |`($x ^$y) =>`(HPow.hPow $x $y)
open scoped RealInnerProductSpace
open BigOperators Finset ENNReal NNReal MeasureTheory
set_option trace.Meta.Tactic.simp.rewrite true
set_option maxHeartbeats 400000/- We defined measures μ and π (ν is considered as the standard Lebesgue measure) along with their densities (finite and non-zero on the entire space)-/variable {d : ℕ}
variable [MeasurableSpace (Vector ℝ d)] [MeasureSpace (Vector ℝ d)] [MeasureSpace ℝ]
variable (μ π ν : Measure (Vector ℝ d)) (dμ dπ : (Vector ℝ d) → ℝ≥0∞)
/- μ << π << ν, they both admit density w.r.t. ν.-/variable (_h1 : absolutely_continuous μ π) (_h2 : absolutely_continuous π ν)
example : absolutely_continuous μ ν := absolutely_continuous_trans _h1 _h2
variable (hμ : is_density μ ν dμ) (hπ : is_density π ν dπ) (mdμ : Measurable dμ) (mdπ : Measurable dπ) (hdμ : ∀x, dμ x ≠0∧ dμ x ≠∞) (hdπ : ∀x, dπ x ≠0∧ dπ x ≠∞)
variable [IsProbabilityMeasure μ] [IsProbabilityMeasure π]
variable (h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet s)
/- We define a RKHS of ((Vector ℝ d) → ℝ) functions.-/variable (H₀ : Set ((Vector ℝ d) → ℝ)) [NormedAddCommGroup ((Vector ℝ d) → ℝ)] [InnerProductSpace ℝ ((Vector ℝ d) → ℝ)]
/- The kernel function -/variable (k : (Vector ℝ d) → (Vector ℝ d) → ℝ) (h_k : (∀ (x : (Vector ℝ d)), k x ∈ H₀) ∧ (∀ (x : (Vector ℝ d)), (fun y ↦ k y x) ∈ H₀))
variable (h_kernel : is_kernel H₀ k) (h_kernel_positive : positive_definite_kernel μ k)
/- We define the product RKHS as a space of function on ℕ → (Vector ℝ d) to ℝ (vector-valued function in our Lean formalism). A function belongs to such a RKHS if f = (f_1, ..., f_d) and ∀ 1 ≤ i ≤ d, fᵢ ∈ H₀. -/variable {H : Set (ℕ → (Vector ℝ d) → ℝ)} [NormedAddCommGroup (ℕ → (Vector ℝ d) → ℝ)] [InnerProductSpace ℝ (ℕ → (Vector ℝ d) → ℝ)]
/-===============================KERNEL STEIN DISCREPANCY===============================-//-Here, we prove that KSD(μ | π) is a valid discrepancy measure, that the Stein log Sobolev inequality holds, and the exponential convergence of SVGD.-//- dk : x ↦ i ↦ y ↦ ∂xⁱ k(x, y) -/variable (dk : (Vector ℝ d) → ℕ → (Vector ℝ d) → ℝ)
/- d_log_π : i ↦ x ↦ ∂xⁱ log (μ(x) / π(x)) -/variable (d_log_π : ℕ → (Vector ℝ d) → ℝ)
/- Definition of the steepest direction ϕ -/variable (ϕ : ℕ → (Vector ℝ d) → ℝ) (hϕ : ϕ ∈ H) (dϕ : ℕ → (Vector ℝ d) → ℝ)
variable (h_is_ϕ : is_phi μ k dk d_log_π ϕ)
/- We will use this assumption only when the function is trivially integrable (e.g. derivative of integrable functions). -/variable (is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable f μ)
/-d_log_π_μ : i ↦ x ↦ ∂xⁱ log (π(x) / μ(x))-/variable (d_log_π_μ : ℕ → (Vector ℝ d) → ℝ)
/-Simple derivative rule: if the derivative is 0 ∀x, then the function is constant.-/variable (hd_log_π_μ : (∀x, ∀i, d_log_π_μ i x =0) → (∃ c, ∀ x, log (dμ x / dπ x) = c))
/-dπ' : i ↦ x ↦ ∂xⁱ π(x)-/variable (dπ' : ℕ → (Vector ℝ d) → ℝ)
/-Simple derivative rule: ∂xⁱ log (π(x)) * π(x) = ∂xⁱ π(x).-/variable (hπ' : ∀x, ∀i, ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x)
variable [Norm (Vector ℝ d)]
/-- Stein class of measure. f is in the Stein class of μ if, ∀i ∈ range (d + 1), lim_(‖x‖ → ∞) μ(x) * ϕ(x)ⁱ = 0.-/def SteinClass (f : ℕ → (Vector ℝ d) → ℝ) (dμ : (Vector ℝ d) → ℝ≥0∞) := ∀ x, tends_to_infty (fun (x : Vector ℝ d) ↦‖x‖) →∀i, ENNReal.toReal (dμ x) * f i x =0/- Kernel Stein Discrepancy-/variable (KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ)
/--KSD(μ | π) = ⟪∇log π/μ, Pμ ∇log π/μ⟫_L²(μ). We assume here that KSD is also equal to ∫ x, ∑ l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ.-/def is_ksd := KSD μ π = (∫ x in Set.univ, (∫ x' in Set.univ, (∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x') ∂μ) ∂μ) ∧ (KSD μ π =∫ x, ∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ)
/- KSD(μ | π) is originally defined as ‖ϕ^⋆‖²_H, it is therefore non-negative.-/variable (ksd_nn : 0≤ KSD μ π)
/- ϕ is in the Stein class of π-/variable (hstein : SteinClass ϕ dπ)
/-- We show that, if ϕ is in the Stein class of π, KSD is a valid discrepancy measure i.e. μ = π ↔ KSD(μ | π) = 0.-/lemma KSD_is_valid_discrepancy (hksd : is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD) : μ = π ↔ KSD μ π =0 :=
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
μ = π ↔ KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
μ = π ↔ KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
-- μ = π ↦ KSD(μ | π) = 0.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π
mp
KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π
mp
KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ =0
-- ∑ i, f i + g i = ∑ i, f i + ∑ i, g i.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ =0
[Meta.Tactic.simp.rewrite] split_sum:1000, ∑ l in Finset.range (d +1),
(d_log_π l x * ϕ l x +
dϕ l x) ==>∑ l in Finset.range (d +1), d_log_π l x * ϕ l x +∑ l in Finset.range (d +1), dϕ l x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x ∂μ =0
-- Split the integral of sum into sum of integral.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x
mp
∫ (x : Vector ℝ d), ∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x
mp
∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x
mp
∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
-- Make the `Set.univ` appears for using the density later.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x
mp
∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x
∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict μ Set.univ ==> μ
[Meta.Tactic.simp.rewrite] @eq_self:1000, ∫ (a : Vector ℝ d), ∑ l in Finset.range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d), ∑ l in Finset.range (d +1), d_log_π l a * ϕ l a ∂μ ==> True
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
-- Replace μ by π in the integration.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂μ =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂π +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂π +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
-- Replace by its density.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂π +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), d_log_π l x * ϕ l x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), d_log_π l x * ϕ l x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
-- Get ENNReal.toReal (dπ x) in the sum (a * ∑ b = ∑ b * a).
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
mp
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), d_log_π l x * ϕ l x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist_sum: ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d +1), f i) * a =∑ i in range (d +1), f i * a
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist_sum: ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d +1), f i) * a =∑ i in range (d +1), f i * a x: Vector ℝ d
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist_sum: ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d +1), f i) * a =∑ i in range (d +1), f i * a x: Vector ℝ d
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist_sum: ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d +1), f i) * a =∑ i in range (d +1), f i * a x: Vector ℝ d
(∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l) * ENNReal.toReal (dπ x) =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist_sum: ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d +1), f i) * a =∑ i in range (d +1), f i * a x: Vector ℝ d
(∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l) * ENNReal.toReal (dπ x) =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
Goals accomplished!🐙
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mp
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), d_log_π l x * ϕ l x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
[Meta.Tactic.simp.rewrite] mul_dist:1000, ENNReal.toReal (dπ x) *∑ l in Finset.range (d +1),
d_log_π l x * ϕ l x ==>∑ l in Finset.range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
-- Make the product ENNReal.toReal (dπ x) * d_log_π i x appears to use the log derivative rule.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
[Meta.Tactic.simp.rewrite] mul_comm:1000, d_log_π x x✝* ϕ x x✝*
ENNReal.toReal (dπ x✝) ==> ENNReal.toReal (dπ x✝) * d_log_π x x✝* ϕ x x✝
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), ENNReal.toReal (dπ x) * d_log_π x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
[Meta.Tactic.simp.rewrite] hπ':1000, ENNReal.toReal (dπ x✝) * d_log_π x x✝==> dπ' x x✝
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
-- Make the `Set.univ` appears to use the density.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict π Set.univ ==> π
[Meta.Tactic.simp.rewrite] @eq_self:1000, ∫ (a : Vector ℝ d), ∑ l in Finset.range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d), ∑ l in Finset.range (d +1), dϕ l a ∂π ==> True
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), dϕ l x ∂ν =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), dϕ l x ∂ν =0
-- Use the integration by parts on the right-hand side integral.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) *∑ l in range (d +1), dϕ l x ∂ν =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +-∫ (x : Vector ℝ d) in Set.univ, ∑ i in range (d +1), dπ' i x * ϕ i x ∂ν =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ = π split_sum: ∀ (x : Vector ℝ d),
∑ l in range (d +1), (d_log_π l x * ϕ l x + dϕ l x) =∑ l in range (d +1), d_log_π l x * ϕ l x +∑ l in range (d +1), dϕ l x h1: Integrable fun x =>∑ l in range (d +1), d_log_π l x * ϕ l x h2: Integrable fun x =>∑ l in range (d +1), dϕ l x int_univ✝: ∫ (a : Vector ℝ d), ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), d_log_π l a * ϕ l a ∂μ mul_dist: ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) *∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l =∑ l in range (d +1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x) mul_comm: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x int_univ: ∫ (a : Vector ℝ d), ∑ l in range (d +1), dϕ l a ∂π =∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d +1), dϕ l a ∂π
mp
∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +-∫ (x : Vector ℝ d) in Set.univ, ∑ i in range (d +1), dπ' i x * ϕ i x ∂ν =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mp
μ = π → KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π =0→ μ = π
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict ν Set.univ ==> ν
[Meta.Tactic.simp.rewrite] @add_right_neg:1000, ∫ (x : Vector ℝ d),
∑ x_1 in Finset.range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν +-∫ (x : Vector ℝ d), ∑ x_1 in Finset.range (d +1), dπ' x_1 x * ϕ x_1 x ∂ν ==>0
[Meta.Tactic.simp.rewrite] @eq_self:1000, 0=0==> True
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mpr
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
μ = π ↔ KSD μ π =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mpr
KSD μ π =0→ μ = π
-- KSD(μ | π) = 0 ↦ μ = π.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: KSD μ π =0
mpr
μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mpr
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: KSD μ π =0
mpr
μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0
mpr
μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0
mpr
μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0
mpr
μ = π
-- We use the fact that the kernel is positive-definite that implies that d_log_π_μ = 0.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0
mpr
μ = π
-- Simple derivative rule: ∂x f x = 0 → f x = c
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 hd_log_π_μ: ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
mpr
μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
mpr
KSD μ π =0→ μ = π
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
mpr.intro
μ = π
-- We show that, since dμ x / dπ x ≠ 0 and finite, dμ x = ENNReal.ofReal (Real.exp c) * dπ x.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
mpr.intro
μ = π
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c x: Vector ℝ d
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c
dμ x / dπ x ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c
dμ x / dπ x ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c
dμ x / dπ x ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_pos: 0< dμ x / dπ x
dμ x / dπ x ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c
dμ x / dπ x ≠0
Goals accomplished!🐙
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0
dμ x / dπ x ≠⊤
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0
dμ x / dπ x ≠⊤
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0
dμ x / dπ x ≠⊤
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝¹: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h✝: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 h: dμ x / dπ x =⊤
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0
dμ x / dπ x ≠⊤
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝¹: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h✝: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 h: dμ x / dπ x =⊤
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝¹: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h✝: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 h: dμ x ≠0∧ dπ x =0∨ dμ x =⊤∧ dπ x ≠⊤
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝¹: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h✝: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 h: dμ x ≠0∧ dπ x =0∨ dμ x =⊤∧ dπ x ≠⊤
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝¹: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h✝: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 h: dμ x ≠0∧ dπ x =0∨ dμ x =⊤∧ dπ x ≠⊤
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0
dμ x / dπ x ≠⊤
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝¹: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h✝: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 h: dμ x ≠0∧ dπ x =0∨ dμ x =⊤∧ dπ x ≠⊤
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hp: dμ x ≠0∧ dπ x =0
inl
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hp: dμ x ≠0∧ dπ x =0
inl
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hp: dμ x ≠0∧ dπ x =0
inl
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hp: dμ x ≠0∧ dπ x =0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hq: dμ x =⊤∧ dπ x ≠⊤
inr
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hql: dμ x =⊤ hqr: dπ x ≠⊤
inr.intro
False
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 hq: dμ x =⊤∧ dπ x ≠⊤
inr
False
Goals accomplished!🐙
Goals accomplished!🐙
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ x: Vector ℝ d h: log (dμ x / dπ x) = c frac_neq_zero: dμ x / dπ x ≠0 frac_finite: dμ x / dπ x ≠⊤ cancel_log_exp: dμ x / dπ x = ENNReal.ofReal (Real.exp c)
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
[Meta.Tactic.simp.rewrite] ← cancel_log_exp:1000, ENNReal.ofReal (Real.exp c) ==> dμ x / dπ x
[Meta.Tactic.simp.rewrite] @ENNReal.div_eq_inv_mul:1000, dμ x / dπ x ==> (dπ x)⁻¹* dμ x
[Meta.Tactic.simp.rewrite] mul_right_comm (dπ x)⁻¹ (dμ x) (dπ x):1000, (dπ x)⁻¹* dμ x * dπ x ==> (dπ x)⁻¹* dπ x * dμ x
[Meta.Tactic.simp.rewrite] ENNReal.inv_mul_cancel (hdπ x).left (hdπ x).right:1000, (dπ x)⁻¹* dπ x ==>1
[Meta.Tactic.simp.rewrite] @one_mul:1000, 1* dμ x ==> dμ x
[Meta.Tactic.simp.rewrite] @eq_self:1000, dμ x = dμ x ==> True
Goals accomplished!🐙
Goals accomplished!🐙
-- We show by cases that ENNReal.ofReal (Real.exp c) = 1. If it is ≠ 1, this implies a contradiction as dμ x = ENNReal.ofReal (Real.exp c) * dπ x and ∫⁻ x, dμ x ∂ν = 1.
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
mpr.intro
μ = π
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ENNReal.ofReal (Real.exp c) =1
pos
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ¬ENNReal.ofReal (Real.exp c) =1
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ENNReal.ofReal (Real.exp c) =1
pos
ENNReal.ofReal (Real.exp c) =1
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ¬ENNReal.ofReal (Real.exp c) =1
neg
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ¬ENNReal.ofReal (Real.exp c) =1
neg
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ¬ENNReal.ofReal (Real.exp c) =1
neg
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ENNReal.ofReal (Real.exp c) ≠1
neg
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ¬ENNReal.ofReal (Real.exp c) =1
neg
ENNReal.ofReal (Real.exp c) =1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ, dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h✝: ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d +1), d_log_π_μ i x * k x x' * d_log_π_μ i x' ∂μ ∂μ =0 d_log_π_μ_eq_0: ∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0 c: ℝ h: ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dμ_propor: ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x hc: ENNReal.ofReal (Real.exp c) ≠1
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0
∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
a ≤1/ (2* (b / (2* a))) * b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
a ≤ (2* (b / (2* a)))⁻¹* b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
a ≤ (2* (b / (2* a)))⁻¹* b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0
∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
a ≤ (2* (b / (2* a)))⁻¹* b
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = (2* (b / (2* a)))⁻¹* b
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = (2* (b * (2* a)⁻¹))⁻¹* b
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b * (2* a)⁻¹))⁻¹* b = (2* b * (2* a)⁻¹)⁻¹* b
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2* (b * (2* a)⁻¹) ==>2* b * (2* a)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, (2* b * (2* a)⁻¹)⁻¹^1==> (2* b * (2* a)⁻¹)⁻¹
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @mul_one:1000, b *1==> b
[Meta.Tactic.simp.rewrite] @add_zero:1000, (2* b * (2* a)⁻¹)⁻¹* b +0==> (2* b * (2* a)⁻¹)⁻¹* b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (2* b * (2* a)⁻¹)⁻¹^1==> (2* b * (2* a)⁻¹)⁻¹
[Meta.Tactic.simp.rewrite] @mul_one:1000, (2* b * (2* a)⁻¹)⁻¹*1==> (2* b * (2* a)⁻¹)⁻¹
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (2* b * (2* a)⁻¹)⁻¹+0==> b * (2* b * (2* a)⁻¹)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, a ^1==> a
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @add_zero:1000, a *2+0==> a *2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, a ^1==> a
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @add_zero:1000, a *2+0==> a *2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (a *2)⁻¹^1==> (a *2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a *2)⁻¹*2) ==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a *2)⁻¹*2+0==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, a ^1==> a
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @add_zero:1000, a *2+0==> a *2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (a *2)⁻¹^1==> (a *2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a *2)⁻¹*2) ==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a *2)⁻¹*2+0==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (b * (a *2)⁻¹*2)⁻¹^1==> (b * (a *2)⁻¹*2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @mul_one:1000, (b * (a *2)⁻¹*2)⁻¹*1==> (b * (a *2)⁻¹*2)⁻¹
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (b * (a *2)⁻¹*2)⁻¹+0==> b * (b * (a *2)⁻¹*2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, a ^1==> a
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @add_zero:1000, a *2+0==> a *2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, a ^1==> a
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @add_zero:1000, a *2+0==> a *2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (a *2)⁻¹^1==> (a *2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a *2)⁻¹*2) ==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a *2)⁻¹*2+0==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, a ^1==> a
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @add_zero:1000, a *2+0==> a *2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (a *2)⁻¹^1==> (a *2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2==>2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a *2)⁻¹*2) ==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a *2)⁻¹*2+0==> b * (a *2)⁻¹*2
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @pow_one:1000, b ^1==> b
[Meta.Tactic.simp.rewrite] @pow_one:1000, (b * (a *2)⁻¹*2)⁻¹^1==> (b * (a *2)⁻¹*2)⁻¹
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1==>1
[Meta.Tactic.simp.rewrite] @mul_one:1000, (b * (a *2)⁻¹*2)⁻¹*1==> (b * (a *2)⁻¹*2)⁻¹
[Meta.Tactic.simp.rewrite] @add_zero:1000, b * (b * (a *2)⁻¹*2)⁻¹+0==> b * (b * (a *2)⁻¹*2)⁻¹
[Meta.Tactic.simp.rewrite] @eq_self:1000, b * (b * (a *2)⁻¹*2)⁻¹= b * (b * (a *2)⁻¹*2)⁻¹==> True
Try this: ring_nf
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* (b / (2* a)))⁻¹* b = a
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* b * (2* a)⁻¹)⁻¹* b = (2*2⁻¹* a⁻¹* b)⁻¹* b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* b * (2* a)⁻¹)⁻¹* b = (2*2⁻¹* a⁻¹* b)⁻¹* b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* b * (2* a)⁻¹)⁻¹* b = (2*2⁻¹* a⁻¹* b)⁻¹* b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* b * (2* a)⁻¹)⁻¹* b = (2*2⁻¹* a⁻¹* b)⁻¹* b
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
(2* b * (2* a)⁻¹)⁻¹* b = (2*2⁻¹* a⁻¹* b)⁻¹* b
Goals accomplished!🐙
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 a, b: ℝ≥0∞ h0a: a ≠0 hta: a ≠⊤ h0b: b ≠0 htb: b ≠⊤ simpl: 1/ (2* (b / (2* a))) = (2* (b / (2* a)))⁻¹
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
ENNReal.ofReal (KSD μ π) ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
ENNReal.ofReal (KSD μ π) ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
ENNReal.ofReal (KSD μ π) ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b KSD_ge_0: 0< KSD μ π
ENNReal.ofReal (KSD μ π) ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
ENNReal.ofReal (KSD μ π) ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b KSD_ge_0: 0< KSD μ π enn_KSD_ge_0: 0< ENNReal.ofReal (KSD μ π)
ENNReal.ofReal (KSD μ π) ≠0
d: ℕ inst✝⁹: MeasurableSpace (Vector ℝ d) inst✝⁸: MeasureSpace (Vector ℝ d) inst✝⁷: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝⁶: IsProbabilityMeasure μ inst✝⁵: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁴: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝²: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD h: μ ≠ π KL_neq_0: KL μ dμ dπ ≠0 calculation: ∀ (a b : ℝ≥0∞), a ≠0→ a ≠⊤→ b ≠0→ b ≠⊤→ a ≤1/ (2* (b / (2* a))) * b
ENNReal.ofReal (KSD μ π) ≠0
Goals accomplished!🐙
Goals accomplished!🐙
-- Use the previous calculation with a := KL(μ || π), b := KSD(μ | π).
Goals accomplished!🐙
Goals accomplished!🐙
Goals accomplished!🐙
Goals accomplished!🐙
/- In this sub-section, we define the flow of measures μ_t: μ_t : ℝ≥0 → Measure (Vector ℝ d) t ↦ T_t#μ, where T_t is the trajectories associated with ϕ(μ_t t), the steepest direction to update μ_t t for minimizing ∂t KL(μ_t t || π). We also define everything that we need to use previous results with each measures given by μ_t.-/variable (μ_t : ℝ≥0→ Measure (Vector ℝ d)) (dμ_t : ℝ≥0→ (Vector ℝ d → ℝ≥0∞)) (hμ_t : ∀ t, is_density (μ_t t) ν (dμ_t t)) (h_prob : ∀ t, IsProbabilityMeasure (μ_t t))
variable (hdμ_t :∀t, ∀ (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤)
/- d_KL_t : t ↦ ∂t KL(μ_t t || π)-/variable (d_KL_t : ℝ≥0→ ℝ)
variable (d_log_μ_t_π : ℝ≥0→ ℕ → (Vector ℝ d) → ℝ)
variable (hd_log_μ_t_π : ∀t, (∀x, ∀i, d_log_μ_t_π t i x =0) → (∃ c, ∀ x, log (dμ_t t x / dπ x) = c))
variable (hkl_eq_t : ∀t, μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0) (hkl_diff_t : ∀t, μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ)
variable (h_kernel_positive_t : ∀t, positive_definite_kernel (μ_t t) k)
variable (is_integrable_H₀_t : ∀t, ∀ (f : Vector ℝ d → ℝ), Integrable f (μ_t t))
variable (ksd_nn_t : ∀t, 0≤ KSD (μ_t t) π)
/- Our definition of the Gronwall's lemma.-/variable [MeasureSpace ℝ≥0] [NormedAddCommGroup ℝ≥0∞] [NormedSpace ℝ ℝ≥0∞] [LocallyFiniteOrder ℝ≥0]
variable (gronwall : ∀ (ψ : ℝ≥0→ ℝ), ∀ t >0, d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) → KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ s in Icc 0 t, ψ s))
/- We assume here that ∂t KL(μ_t t || π) = - KSD(μ_t t | π) (showed in the paper).-/variable (dkl_ksd : ∀t, d_KL_t t ≤- KSD (μ_t t) π)
/- As showed in the paper (and reminded above), t ↦ KL(μ_t t || π) is decreasing.-/variable (kl_decreasing : ∀t, ∀t', t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ) (kl_finite : ∀t, KL (μ_t t) (dμ_t t) dπ ≠∞)
/--If t ↦ KL(μ_t t || π) is bounded from below by a strictly positive constant, it means that μ_t cannot be made arbitrary close to π and thus that t ↦ KSD(μ_t | π) can be bounded by a striclty positive constant. Admitted here; we plan on formally show it in the future.-/lemma
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0
a * (2* c) ≤ (2* c)⁻¹* b * (2* c)
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0
a * (2* c) ≤ (2* c)⁻¹* b * (2* c)
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0
a * (2* c) ≤ (2* c)⁻¹* b * (2* c)
Goals accomplished!🐙
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c)
(2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c))
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c) ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c))
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c) ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c))
(2* c)⁻¹* (2* c) =1
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c) ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b * ((2* c)⁻¹* (2* c)) ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c) ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b *1 ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ (2* c)⁻¹* b * (2* c) ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
ENNReal.toReal (a * (2* c)) ≤ ENNReal.toReal b
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
ENNReal.toReal (a * (2* c)) ≤ ENNReal.toReal b
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t: (2* c)⁻¹* (2* c) =1
ENNReal.toReal (a * (2* c)) ≤ ENNReal.toReal b
Goals accomplished!🐙
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal (a * (2* c)) ≤ ENNReal.toReal b
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal (a * (2* c)) ≤ ENNReal.toReal b
ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
[Meta.Tactic.simp.rewrite] @ENNReal.toReal_mul:1000, ENNReal.toReal
(a * (2* c)) ==> ENNReal.toReal a * ENNReal.toReal (2* c)
[Meta.Tactic.simp.rewrite] @ENNReal.toReal_mul:1000, ENNReal.toReal (2* c) ==> ENNReal.toReal 2* ENNReal.toReal c
[Meta.Tactic.simp.rewrite] ENNReal.toReal_ofNat:1000, ENNReal.toReal 2==>2
[Meta.Tactic.simp.rewrite] @eq_self:1000, ENNReal.toReal a * (2* ENNReal.toReal c) =
ENNReal.toReal a * (2* ENNReal.toReal c) ==> True
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal (a * (2* c)) ≤ ENNReal.toReal b tt: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal (2* c) ≤ ENNReal.toReal b tt: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal (2* c) ≤ ENNReal.toReal b tt: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal (2* c) ≤ ENNReal.toReal b tt: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal (2* c) ≤ ENNReal.toReal b tt: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal (2* c) ≤ ENNReal.toReal b tt: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c)
ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
[Meta.Tactic.simp.rewrite] @ENNReal.toReal_mul:1000, ENNReal.toReal (2* c) ==> ENNReal.toReal 2* ENNReal.toReal c
[Meta.Tactic.simp.rewrite] ENNReal.toReal_ofNat:1000, ENNReal.toReal 2==>2
[Meta.Tactic.simp.rewrite] @eq_self:1000, 2* ENNReal.toReal c =2* ENNReal.toReal c ==> True
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝¹: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal (2* c) ≤ ENNReal.toReal b tt✝: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝¹: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) ≤ ENNReal.toReal b tt✝: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝¹: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) ≤ ENNReal.toReal b tt✝: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝¹: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) ≤ ENNReal.toReal b tt✝: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝¹: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) ≤ ENNReal.toReal b tt✝: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝¹: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) ≤ ENNReal.toReal b tt✝: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c
ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝²: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) ≤ ENNReal.toReal b tt✝¹: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝²: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝¹: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝²: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝¹: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝²: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝¹: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c)
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝¹: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝: (2* c)⁻¹* (2* c) =1 t: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c)
-(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c) =-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝²: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝¹: (2* c)⁻¹* (2* c) =1 t✝: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c) t: -(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c) =-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝²: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝¹: (2* c)⁻¹* (2* c) =1 t✝: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a t: -(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c) =-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝²: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝¹: (2* c)⁻¹* (2* c) =1 t✝: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a t: -(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c) =-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π) a, b, c: ℝ≥0∞ htb: b ≠⊤ h0c: c ≠0 htc: c ≠⊤ h: a ≤ (2* c)⁻¹* b t✝²: 1/ (2* c) * b = (2* c)⁻¹* b finite: 2* c ≠⊤ n_zero: 2* c ≠0 tt✝³: a * (2* c) ≤ b ttt: (2* c)⁻¹* b * (2* c) = b * ((2* c)⁻¹* (2* c)) t✝¹: (2* c)⁻¹* (2* c) =1 t✝: ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c ≤ ENNReal.toReal b tt✝²: ENNReal.toReal (a * (2* c)) = ENNReal.toReal a * ENNReal.toReal (2* c) tt✝¹: ENNReal.toReal (2* c) = ENNReal.toReal 2* ENNReal.toReal c tt✝: ENNReal.toReal a * (ENNReal.toReal 2* ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c tt: -ENNReal.toReal b ≤-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a t: -(ENNReal.toReal a * ENNReal.toReal 2* ENNReal.toReal c) =-ENNReal.toReal 2* ENNReal.toReal c * ENNReal.toReal a
-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ (a b c : ℝ≥0∞),
b ≠⊤→ c ≠0→ c ≠⊤→ a ≤1/ (2* c) * b →-ENNReal.toReal b ≤-2* ENNReal.toReal c * ENNReal.toReal a
Goals accomplished!🐙
Goals accomplished!🐙
d: ℕ inst✝¹³: MeasurableSpace (Vector ℝ d) inst✝¹²: MeasureSpace (Vector ℝ d) inst✝¹¹: MeasureSpace ℝ μ, π, ν: Measure (Vector ℝ d) dμ, dπ: Vector ℝ d → ℝ≥0∞ _h1: absolutely_continuous μ π _h2: absolutely_continuous π ν hμ: is_density μ ν dμ hπ: is_density π ν dπ mdμ: Measurable dμ mdπ: Measurable dπ hdμ: ∀ (x : Vector ℝ d), dμ x ≠0∧ dμ x ≠⊤ hdπ: ∀ (x : Vector ℝ d), dπ x ≠0∧ dπ x ≠⊤ inst✝¹⁰: IsProbabilityMeasure μ inst✝⁹: IsProbabilityMeasure π h_m_set: ∀ (s : Set (Vector ℝ d)), MeasurableSet s H₀: Set (Vector ℝ d → ℝ) inst✝⁸: NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷: InnerProductSpace ℝ (Vector ℝ d → ℝ) k: Vector ℝ d → Vector ℝ d → ℝ h_k: (∀ (x : Vector ℝ d), k x ∈ H₀) ∧∀ (x : Vector ℝ d), (fun y => k y x) ∈ H₀ h_kernel: is_kernel H₀ k h_kernel_positive: positive_definite_kernel μ k H: Set (ℕ → Vector ℝ d → ℝ) inst✝⁶: NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵: InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk: Vector ℝ d → ℕ → Vector ℝ d → ℝ d_log_π, ϕ: ℕ → Vector ℝ d → ℝ hϕ: ϕ ∈ H dϕ: ℕ → Vector ℝ d → ℝ h_is_ϕ: is_phi μ k dk d_log_π ϕ is_integrable_H₀: ∀ (f : Vector ℝ d → ℝ), Integrable f d_log_π_μ: ℕ → Vector ℝ d → ℝ hd_log_π_μ: (∀ (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ': ℕ → Vector ℝ d → ℝ hπ': ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x inst✝⁴: Norm (Vector ℝ d) KSD: Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn: 0≤ KSD μ π hstein: SteinClass ϕ dπ hkl_eq: μ = π → KL μ dμ dπ =0 hkl_diff: μ ≠ π →0< KL μ dμ dπ μ_t: ℝ≥0→ Measure (Vector ℝ d) dμ_t: ℝ≥0→ Vector ℝ d → ℝ≥0∞ hμ_t: ∀ (t : ℝ≥0), is_density (μ_t t) ν (dμ_t t) h_prob: ∀ (t : ℝ≥0), IsProbabilityMeasure (μ_t t) hdμ_t: ∀ (t : ℝ≥0) (x : Vector ℝ d), dμ_t t x ≠0∧ dμ_t t x ≠⊤ d_KL_t: ℝ≥0→ ℝ d_log_μ_t_π: ℝ≥0→ ℕ → Vector ℝ d → ℝ hd_log_μ_t_π: ∀ (t : ℝ≥0), (∀ (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x =0) →∃ c, ∀ (x : Vector ℝ d), log (dμ_t t x / dπ x) = c hkl_eq_t: ∀ (t : ℝ≥0), μ_t t = π → KL (μ_t t) (dμ_t t) dπ =0 hkl_diff_t: ∀ (t : ℝ≥0), μ_t t ≠ π →0< KL (μ_t t) (dμ_t t) dπ h_kernel_positive_t: ∀ (t : ℝ≥0), positive_definite_kernel (μ_t t) k is_integrable_H₀_t: ∀ (t : ℝ≥0) (f : Vector ℝ d → ℝ), Integrable f ksd_nn_t: ∀ (t : ℝ≥0), 0≤ KSD (μ_t t) π inst✝³: MeasureSpace ℝ≥0 inst✝²: NormedAddCommGroup ℝ≥0∞ inst✝¹: NormedSpace ℝ ℝ≥0∞ inst✝: LocallyFiniteOrder ℝ≥0 gronwall: ∀ (ψ : ℝ≥0→ ℝ),
∀ t >0,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0) (dμ_t 0) dπ * exp (∫ (s : ℝ≥0) in↑(Icc 0 t), ψ s) dkl_ksd: ∀ (t : ℝ≥0), d_KL_t t ≤-KSD (μ_t t) π kl_decreasing: ∀ (t t' : ℝ≥0), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ kl_finite: ∀ (t : ℝ≥0), KL (μ_t t) (dμ_t t) dπ ≠⊤ log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ ≤1/ (2* (ENNReal.ofReal (KSD (μ_t t) π) / (2* KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) l: ℝ≥0∞ lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos: 0≤ l KL_bounded: ∀ (x : ℝ≥0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x hl: 0≠ l lim_nn: 0< l γ: ℝ≥0∞ KSD_bounded: ∀ (t : ℝ≥0), γ < ENNReal.ofReal (KSD (μ_t t) π) γ_nn: 0< γ γ_finite: γ ≠⊤ gamma_star: ∀ (t : ℝ≥0),
γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) t) gamma_star_neq: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠0 gamma_star_finite: γ / (2* KL (μ_t 0) (dμ_t 0) dπ) ≠⊤ bounded_log_sobolev: ∀ (t : ℝ≥0),
KL (μ_t t) (dμ_t t) dπ <1/ (2* (γ / (2* (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)